Subgroups — C₂²×#15688969 #26374448

Order 12032.

5150 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×Dic₄₇	C₂	1	2	yes	maximal
2	C₂×#15688969	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄₇⋊#95	C₂	1	4	yes	maximal
4	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₄₇	C₂²	1	4	yes	—
6	C₂²×C₄×Dic₄₇	C₄	1	32	yes	complemented
7	C₂²×C₄×Dic₄₇	C₂²	1	24	yes	—
8	#15688969	C₂²	1	64	yes	direct product factor
9	C₂×C₄₇⋊#95	C₂²	1	48	yes	—
10	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₄₇	C₄	1	16	yes	complemented
13	C₂²×C₄²×C₄₇	C₂²	1	12	yes	—
14	C₂³×Dic₄₇	—	2	32	no	—
15	C₂³×Dic₄₇	C₂×C₄	1	32	yes	—
16	C₂³×Dic₄₇	C₂³	1	24	yes	—
17	C₂×C₄×Dic₄₇	C₂×C₄	1	192	yes	complemented
18	C₂×C₄×Dic₄₇	C₂³	1	32	yes	—
19	C₄₇⋊#95	C₂³	1	64	yes	—
20	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₄₇	—	2	16	no	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	96	yes	complemented
28	C₂×C₄²×C₄₇	C₂³	1	16	yes	—
29	C₂⁴×C₄₇	C₂²×C₄	1	8	yes	—
30	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
31	C₂⁴×C₄₇	#41	1	16	yes	—
32	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
33	C₂²×C₄×C₄₇	—	2	64	no	—
34	C₂²×C₄×C₄₇	—	2	48	no	—
35	C₂²×C₄×C₄₇	C₄²	1	64	yes	—
36	C₂²×C₄×C₄₇	C₂²×C₄	1	48	yes	—
37	C₂²×C₄×C₄₇	C₂²×C₄	1	48	yes	—
38	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
39	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
40	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
41	C₂²×Dic₄₇	—	2	128	no	—
42	C₂²×Dic₄₇	—	2	96	no	—
43	C₂²×Dic₄₇	C₂²×C₄	1	96	yes	—
44	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
45	C₄²×C₄₇	C₂²×C₄	1	64	yes	complemented
46	C₄×Dic₄₇	C₂²×C₄	1	128	yes	complemented
47	C₂³×C₄₇	C₂×C₄²	1	16	yes	—
48	C₂³×C₄₇	#117	1	64	yes	—
49	C₂³×C₄₇	#117	1	32	yes	—
50	C₂³×C₄₇	C₂³×C₄	1	12	yes	—
51	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
52	C₂³×C₄₇	C₂×#41	1	24	yes	—
53	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	96	no	—
55	C₂×C₄×C₄₇	—	2	16	no	—
56	C₂×C₄×C₄₇	C₂×C₄²	1	96	yes	—
57	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
58	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
59	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
60	C₂×Dic₄₇	—	2	192	no	—
61	C₂×Dic₄₇	—	2	32	no	—
62	C₂×Dic₄₇	C₂³×C₄	1	32	yes	—
63	C₂²×#376	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	#376	1	64	yes	complemented
65	C₂²×C₄₇	C₂²×C₄²	1	12	yes	—
66	C₂²×C₄₇	C₂×#117	1	48	yes	—
67	C₂²×C₄₇	C₂×#117	1	24	yes	—
68	C₂²×C₄₇	C₂⁴×C₄	1	2	yes	—
69	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
70	C₂²×C₄₇	C₂²×#41	1	4	yes	—
71	C₄×C₄₇	—	2	16	no	—
72	C₄×C₄₇	C₂²×C₄²	1	16	yes	—
73	Dic₄₇	—	2	32	no	—
74	C₂³×C₄²	—	47	2	no	—
75	C₂³×C₄²	D₄₇	1	1	yes	characteristic
76	C₂×#376	—	47	24	no	—
77	C₂²×#95	—	47	4	no	—
78	C₂×C₄₇	C₂×#376	1	24	yes	complemented
79	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
80	C₂×C₄₇	C₂²×#117	1	4	yes	—
81	C₂×C₄₇	C₂²×#117	1	2	yes	—
82	C₂⁴×C₄	—	47	4	no	—
83	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
84	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
85	C₂²×C₄²	—	47	32	no	—
86	C₂²×C₄²	—	47	24	no	—
87	C₂²×C₄²	Dic₄₇	1	16	yes	complemented
88	C₂²×C₄²	C₂×D₄₇	1	12	yes	—
89	#376	—	47	64	no	—
90	C₂×#95	—	47	48	no	—
91	C₄₇	C₂²×#376	1	1	yes	characteristic, complemented, Sylow
92	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
93	C₂³×C₄	—	2	16	no	—
94	C₂³×C₄	—	47	32	no	—
95	C₂³×C₄	—	47	24	no	—
96	C₂³×C₄	—	94	32	no	—
97	C₂³×C₄	C₄×D₄₇	1	16	yes	—
98	C₂³×C₄	C₂×Dic₄₇	1	16	yes	—
99	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
100	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
101	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
102	C₂×C₄²	—	47	192	no	—
103	C₂×C₄²	—	47	32	no	—
104	C₂×C₄²	C₂×Dic₄₇	1	96	yes	complemented
105	C₂×C₄²	C₂²×D₄₇	1	16	yes	—
106	#95	—	47	64	no	—
107	C₂⁴	C₂×C₄×D₄₇	1	8	yes	—
108	C₂⁴	#10625350	1	16	yes	—
109	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
110	C₂⁴	C₂³×D₄₇	1	3	yes	—
111	C₂²×C₄	—	2	64	no	—
112	C₂²×C₄	—	2	48	no	—
113	C₂²×C₄	—	47	96	no	—
114	C₂²×C₄	—	47	16	no	—
115	C₂²×C₄	—	94	128	no	—
116	C₂²×C₄	—	94	96	no	—
117	C₂²×C₄	C₄×Dic₄₇	1	64	yes	—
118	C₂²×C₄	C₂×C₄×D₄₇	1	48	yes	—
119	C₂²×C₄	C₂²×Dic₄₇	1	48	yes	—
120	C₂²×C₄	C₂²×Dic₄₇	1	24	yes	—
121	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
122	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
123	C₄²	—	47	128	no	—
124	C₄²	C₂²×Dic₄₇	1	64	yes	complemented
125	C₂³	C₂×C₄×Dic₄₇	1	16	yes	—
126	C₂³	#13093981	1	64	yes	—
127	C₂³	#13093983	1	32	yes	—
128	C₂³	C₂²×C₄×D₄₇	1	12	yes	—
129	C₂³	C₂³×Dic₄₇	1	6	yes	—
130	C₂³	C₂×#10625350	1	24	yes	—
131	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
132	C₂×C₄	—	2	96	no	—
133	C₂×C₄	—	2	16	no	—
134	C₂×C₄	—	47	32	no	—
135	C₂×C₄	—	94	192	no	—
136	C₂×C₄	—	94	32	no	—
137	C₂×C₄	C₂×C₄×Dic₄₇	1	96	yes	—
138	C₂×C₄	C₂²×C₄×D₄₇	1	16	yes	—
139	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
140	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
141	C₂²	#15688969	1	64	yes	direct product factor
142	C₂²	C₂²×C₄×Dic₄₇	1	12	yes	—
143	C₂²	C₂×#13093981	1	48	yes	—
144	C₂²	C₂×#13093983	1	24	yes	—
145	C₂²	C₂³×C₄×D₄₇	1	2	yes	—
146	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
147	C₂²	C₂²×#10625350	1	4	yes	—
148	C₄	—	2	16	no	—
149	C₄	—	94	32	no	—
150	C₄	C₂²×C₄×Dic₄₇	1	16	yes	—
151	C₂	C₂×#15688969	1	24	yes	direct product factor
152	C₂	C₂³×C₄×Dic₄₇	1	1	yes	characteristic
153	C₂	C₂²×#13093981	1	4	yes	—
154	C₂	C₂²×#13093983	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂³×C₄²×C₄₇	C₂
socle, prime radical	C₂⁵×C₄₇	C₂³
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂³×C₄²	D₄₇
hypocenter	C₄₇	C₂²×#376
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#15688969	C₂	direct (N × H)
6. C₂²×C₄×Dic₄₇	C₄	semidirect
8. #15688969	C₂²	direct (N × H)
12. C₂²×C₄²×C₄₇	C₄	semidirect
17. C₂×C₄×Dic₄₇	C₂×C₄	semidirect
27. C₂×C₄²×C₄₇	C₂×C₄	semidirect
45. C₄²×C₄₇	C₂²×C₄	semidirect
46. C₄×Dic₄₇	C₂²×C₄	semidirect
64. C₂²×C₄₇	#376	semidirect
78. C₂×C₄₇	C₂×#376	semidirect
87. C₂²×C₄²	Dic₄₇	semidirect
91. C₄₇	C₂²×#376	semidirect
104. C₂×C₄²	C₂×Dic₄₇	semidirect
124. C₄²	C₂²×Dic₄₇	semidirect
141. C₂²	#15688969	direct (N × H)
151. C₂	C₂×#15688969	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₄×Dic₄₇	2
C₂×#15688969	2
C₂²×C₄₇⋊#95	2
C₂³×C₄²×C₄₇	2
C₂²×#376	47

Subgroups — C22×#15688969 #26374448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688969 #26374448